Chop the spinach or silverbeet into 2cm strips and steam until it wilts then squeeze out excess moisture. Set aside in a colander.
Sprinkle all the cheese across one side of each tortilla. Place tortillas - cheese side up - under the grill, and grill 5 minutes, or until cheese is melted.
Melt the butter in a frypan over medium heat. Stir in garlic and mushrooms, and cook about 5 minutes. Mix in spinach, and continue cooking 5 minutes. Place an equal amount of the mixture on the cheese side of each tortilla. Fold tortillas in half over the filling.
Heat oil in a separate frypan over medium heat. Place quesadillas in the frypan one at a time, and cook 3 minutes on each side, until golden brown. Cut each quesadilla into 4 wedges to serve.